Graphs with intrinsic s3 convexities

Separation properties for some intrinsic convexities of graphs are investigated. The most natural convexities defined on a graph are the induced path convexity and the geodesic convexity. A set A of vertices is convex with respect to the former convexity if A contains every induced path connecting two vertices of A. In particular, a characterization of those graphs is given in which all such convex sets are the intersections of halfspaces (i.e., convex sets with convex complements).     

Properties of the Distance Function to Strongly and Weakly Convex Sets in a Nonsymmetrical Space

We consider the distance function (DF), given by the caliber (the Minkowski gauge function) of a convex body, from a point to strictly, strongly, and weakly convex sets in an arbitrary Hilbert space. Some properties of the caliber of a strongly convex set and the conditions for obtaining a strict, strong, or weak convexity of Lebesgue sets for the distance function are established in accordance with the requirements for the set, the caliber of which specifies the distance function, and the set to which the distance is measured. The corresponding inequalities are obtained that reflect the behavior of the distance function on segments and allow comparing it with strictly, strongly, or weakly convex functions.     

Relations between the convexity of a set and the differentiability of its support function

It is known that, in finite dimensions, the support function of a compact convex set with nonempty interior is differentiable excepting the origin if and only if the set is strictly convex. In this paper, we realize a thorough study of the relations between the differentiability of the support function on the interior of its domain and the convexity of the set, mainly for unbounded sets. Then, we revisit some results related to the differentiability of the cost function associated to a production function.     

Book review of Israel Gohberg,Seymour Goldberg,Marinus A. Kaashoek,Basic Classes of Linear Operators,Birkhäuser Verlag,Basel–Boston–Berlin 2003, 423 p. ISBN 3-7643-6930-2

In projective space three notions of convexity (weak convexity, strong convexity, p-convexity) are regarded systematically. Since these notions are defined only by incidence relations, there can be introduced dual notions. We consider relations.between the introduced notions and the most essential properties of convex sets. To all assertions. can be formulated dual assertions, too. The most important theorems given by Fenchel can be generalised. The property of a point set (a set of hyperplanes) to be strongly convex or p-convex, respectively, is invariant with respect to correlations.     

Objective function and logarithmic barrier function properties in convex programming: level sets,solution attainment and strict convexity *

We give several results, some new and some old, but apparently overlooked, that provide useful characterizations of barrier functions and their relationship to problem function properties. In particular, we show that level sets of a barrier function are bounded if and only if feasible level sets of the objective function are bounded and we obtain conditions that imply solution existence, strict convexity or a positive definite Hessian of a barrier function. Attention is focused on convex programs and the logarithmic barrier function. Such results suggest that it would seem possible to extend many recent complexity results by relaxing feasible set compactness to the feasible objective function level set boundedness assumption.     

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.     

DEA with efficiency classification preserving conditional convexity

We propose to relax the standard convexity property used in Data Envelopment Analysis (DEA) by imposing additional qualifications for feasibility of convex combinations. We specifically focus on a condition that preserves the Koopmans efficiency classification. This yields an efficiency classification preserving conditional convexity property, which is implied by both monotonicity and convexity, but not conversely. Substituting convexity by conditional convexity, we construct various empirical DEA approximations as the minimal sets that contain all DMUs and are consistent with the imposed production assumptions. Imposing an additional disjunctive constraint to standard convex DEA formulations can enforce conditional convexity. Computation of efficiency measures relative to conditionally convex production set can be performed through Disjunctive Programming (DP).     

Strong convexity of NTU games

In this article, we provide an alternative notion of NTU convexity: strong ordinal convexity. We show that if a game is strongly ordinal convex, then any marginal vector is in the core, and any marginal payoff is increasing. Some economic examples satisfy this convexity.     

On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in Real Hilbert Spaces

This work focuses on convergence analysis of the projected gradient method for solving constrained convex minimization problems in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo line search converges weakly to a solution of the considered convex optimization problem. Weak convergence is established by assuming convexity and Gateaux differentiability of the objective function, whose Gateaux derivative is supposed to be uniformly continuous on bounded sets. Furthermore, we propose some modifications in the classical projected gradient method in order to obtain strong convergence. The new variant has the following desirable properties: the sequence of generated points is entirely contained in a ball with diameter equal to the distance between the initial point and the solution set, and the whole sequence converges strongly to the solution of the problem that lies closest to the initial iterate. Convergence analysis of both methods is presented without Lipschitz continuity assumption.     

Strong hypercomplex convexity

The concept of a hypercomplex convex set is introduced. The properties of strongly hypercomplex convex sets are considered. A theorem is presented on strong hypercomplex convexity. A hypercomplex version of the geometric form of the Hahn-Banach Theorem is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 182–187, February, 1990.     

Weakly Convex Sets and Their Properties

In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game.     

Strongly extreme points in Banach spaces

In this paper some properties of a special type of boundary point of convex sets in Banach spaces are studied. Specifically, a strongly extreme point x of a convex set S is a point of S such that for each real number r>0, segments of length 2r and centered x are not uniformly closer to S than some positive number d(x,r). Results are obtained comparing the notion of strongly extreme point to other known types of special boundary points of convex sets. Using the notion of strongly extreme point, a convexity condition is defined on the norm of the space under consideration, and this convexity condition makes possible a unified treatment of some previously studied convexity conditions. In addition, a sufficient condition is given on the norm of a separable conjugate space for every extreme point of the unit ball to be strongly extreme.     

Nondifferentiable reverse convex programs and facetial convexity cuts via a disjunctive characterization

Disjunctive Programs can often be transcribed as reverse convex constrained problems with nondifferentiable constraints and unbounded feasible regions. We consider this general class of nonconvex programs, called Reverse Convex Programs (RCP), and show that under quite general conditions, the closure of the convex hull of the feasible region is polyhedral. This development is then pursued from a more constructive standpoint, in that, for certain special reverse convex sets, we specify a finite linear disjunction whose closed convex hull coincides with that of the special reverse convex set. When interpreted in the context of convexity/intersection cuts, this provides the capability of generating any (negative edge extension) facet cut. Although this characterization is more clarifying than computationally oriented, our development shows that if certain bounds are available, then convexity/intersection cuts can be strengthened relatively inexpensively.     

Regularized Newton method for unconstrained convex optimization

We introduce the regularized Newton method (rnm) for unconstrained convex optimization. For any convex function, with a bounded optimal set, the rnm generates a sequence that converges to the optimal set from any starting point. Moreover the rnm requires neither strong convexity nor smoothness properties in the entire space. If the function is strongly convex and smooth enough in the neighborhood of the solution then the rnm sequence converges to the unique solution with asymptotic quadratic rate. We characterized the neighborhood of the solution where the quadratic rate occurs.     

Some properties of convex hulls of integer points contained in general convex sets

In this paper, we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223–225, 1974). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron.     

Peaceman–Rachford splitting for a class of nonconvex optimization problems

We study the applicability of the Peaceman–Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence generated. We then discuss one way to split the objective so that the proposed method can be suitably applied to solving optimization problems with a coercive objective that is the sum of a (not necessarily strongly) convex Lipschitz differentiable function and a proper closed function; this setting covers a large class of nonconvex feasibility problems and constrained least squares problems. Finally, we illustrate the proposed algorithm numerically.     

The e-support function of an e-convex set and conjugacy for e-convex functions

A subset of a locally convex space is called e-convex if it is the intersection of a family of open halfspaces. An extended real-valued function on such a space is called e-convex if its epigraph is e-convex. In this paper we introduce a suitable support function for e-convex sets as well as a conjugation scheme for e-convex functions.     

Dominating Sets for Convex Functions with Some Applications

A number of optimization methods require as a first step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity. In this paper, we address the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an infinite number of) convex functions, and we show how to obtain a convex dominating set in terms of dominating sets of simpler problems. The applicability of the results obtained is illustrated with the statement of new localization results in the fields of linear regression and location.     

Ball and spindle convexity with respect to a convex body

Let ${C \subset \mathbb{R}^n}$ be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either ${\emptyset}$ , or ${\mathbb{R}^n}$ . The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces.     

Remarks on strongly convex functions

Some properties of strongly convex functions are presented. A characterization of pairs of functions that can be separated by a strongly convex function and a Hyers?CUlam stability result for strongly convex functions are given. An integral Jensen-type inequality and a Hermite?CHadamard-type inequality for strongly convex functions are obtained. Finally, a relationship between strong convexity and generalized convexity in the sense of Beckenbach is shown.